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We obtain novel closed form solutions to the Friedmann equation forcosmological models containing a component whose equation of state is that ofradiation $(w=1/3)$ at early times and that of cold pressureless matter $(w=0)$at late times. The equation of state smoothly transitions from the early tolate-time behavior and exactly describes the evolution of a species with aDirac Delta function distribution in momentum magnitudes $|\vec{p}_0|$ (i.e.all particles have the same $|\vec{p}_0|$). Such a component, here termed "hotmatter", is an approximate model for both neutrinos and warm dark matter. Weconsider it alone and in combination with cold matter and with radiation, alsoobtaining closed-form solutions for the growth of super-horizon perturbationsin each case. The idealized model recovers $t(a)$ to better than $1.5\%$accuracy for all $a$ relative to a Fermi-Dirac distribution (as describesneutrinos). We conclude by adding the second moment of the distribution to ourexact solution and then generalizing to include all moments of an arbitrarymomentum distribution in a closed form solution.

Too hot to handle? Analytic solutions for massive neutrino or warm dark matter cosmologies